[PDF] - Completelymixed lake or CSTR Often useful to assume perfect mixing PDF Document

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CEE 572 Lecture #4 9/21/2017 1

Lecture #4 (mass balance, loadings & steady state solutions) Chapra L3

David A. Reckhow CEE 577 #4 1

Updated: 21 September 2017

Print version

Completely‐mixed lake or CSTR

 Often useful to assume perfect mixing

 same concentration throughout system

David A. Reckhow CEE 577 #4 2

Accumulation loading

utflow

reaction settling    

C V

Cin Q C Q

settling reaction Loading Outflow

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CEE 572 Lecture #4 9/21/2017 2

Accumulation

David A. Reckhow CEE 577 #4 3

Accumulation M t   

c M V M Vc   Accumulation Vc t   

and And if volume is constant:

Accumulation V c t V dc dt    

So:

C V

Equals zero at steady state

Loading

 Point Sources

 Municipal Wastewater  Industrial Wastewater  Tributaries

 Non‐point sources

 agricultural  silvicultural  atmospheric  urban & suburban runoff  groundwater

David A. Reckhow CEE 577 #4 4

Diffuse origin more transient

ften dependent on precipitation

Well defined origin easily measured more constant

Loading W t Qc t

in

  ( ) ( )

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CEE 572 Lecture #4 9/21/2017 3

Reported Values Of Selected Waste Input Parameters In The United States

David A. Reckhow CEE 577 #4 5

Variable Unitsa Municipal Influentb CSOc Urban Runoffd Agriculture

(lb/mi2-d) e

Forest

(lb/mi2-d)e

Atmosphere

(lb/mi2-day)f

Average daily flow gcd 125 Total suspended solids mg/L 300 410 610 2500 400 CBOD5

g

mg/L 180 170 27 40 8 CBODU

g

mg/L 220 240 NBOD

g

mg/L 220 290 Total nitrogen mg-N/L 50 9 2.3 15 4 8.9-18.9 Total phosphorus mg-P/L 10 3 0.5 1.0 0.3 0.13-1.3 Total coliforms 10

6/100

mL 30 6 0.3 Cadmium g/L 1.2 10 13 0.015 Lead g/L 22 190 280 1.3 Chromium g/L 42 190 22 0.088 Copper g/L 159 460 110 Zinc g/L 241 660 500 1.8 Total PCB g/L 0.9 0.3

0.002-0.02

(Table 1.3 from Thomann & Mueller)

Footnotes for T&M Table 1.3

David A. Reckhow CEE 577 #4 6

aUnits apply to municipal, CSO (combined sewer overflow), and

urban runoff sources; gcd = gallons per capita per day.

bThomann (1972); heavy metals and PCB, HydroQual (1982). cThomann (1972); total coli, Tetra Tech, (1977); heavy metals Di

Toro et al. (1978): PCB. Hydroscience (1978).

dTetra Tech (1977): heavy metals, Di Toro et al. (1978). eHydroscience (1976a). fNitrogen and phosphorus, Tetra Tech (1982): heavy metals and

PC13, HydroQual (1982).

gCBOD5 = 5 day carbonaceous biochemical oxygen demand

(CBOD); CBODU = ultimate CBOD; NBOD = nitrogenous BOD.

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CEE 572 Lecture #4 9/21/2017 4

Loading: Flow as a function of precipitation

 Non point sources are difficult to characterize

 Empirical approach: export coefficients (see Table 3.1 in

T&M)

 Mechanistic approach: relate to meteorology, topology,

etc.

 Flow: use the rational formula: QR = cIA

David A. Reckhow CEE 577 #4 7

Runoff flow [L3/T] Runoff coefficient

0.1-0.3 for rural areas (1 person/acre) 0.7-0.9 for heavy commercial areas

Rainfall Intensity [L/T] Drainage Area [L2] Note: 1 acre-in/hr 1 cfs

Runoff: Contrasting approaches

 Lumped model

 Empirical  Built on a single rainfall intensity from rain gage data

 Distributed model

 Mechanistic  Built on radar data for rainfall

 Spatial & temporal resolution

 Combine with overland flow models

 Many computer codes

 CASC2D, CUHP, CUHP/SWMM, DR3M, HEC‐1, HSPF, PSRM,

SWMM, TR20

David A. Reckhow CEE 577 #4 8

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CEE 572 Lecture #4 9/21/2017 5

Loading: conc. as a function of flow

 It is common for pollutant concentrations from

uncontrolled sources (e.g. tributaries) to be correlated with flow

David A. Reckhow CEE 577 #4 9

1 10 100 1000 1 10 100 1000 Flow (cfs) Concentration (mg/L)

 establish a log-log relationship  c=aQb Log(C) = log(a) + b*log(Q)

Loading Example: #3.1 from T&M

 Data: Runoff from 100 mi2 of agricultural lands drains to a point in a river where a city of 100,000 people is located. The city has a land area of 10 mi2 and its sanitary sewers are separated from its storm drains. A sewage treatment plant discharges to the river immediately downstream of the

city. The area receives an annual rainfall of 30 in. of which 30% runs off

the agricultural lands and 50% drains off the more impervious city area.  Problem: Using the loading data from Table 1.3 and the residual fractions cited in the table below, compare the contributions of the atmospheric, agricultural and urban sources to annual average values of flow, CBOD5, total coliform bacteria, and lead in the river. Neglect any decay mechanisms for all parameters.

David A. Reckhow CEE 577 #4 10 (at) (ag) (ur) Wastewater Treatment Plant Item Atmospheric Agricultural Urban Runoff Influent

Resid. Fract.

Fow 30% precip. 50% precip. 125 gcd 1.00 CBOD5 40 lb/mi2-d 27 mg/L 180 mg/L 0.15 Total coliform 100/100 mL 3x105/100mL 3x106/100mL 0.0001 Lead 1.3 lb/mi2-d 280 g/L 22 g/L 0.05

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CEE 572 Lecture #4 9/21/2017 6

Solution to loading problem

 Flow contributions

David A. Reckhow CEE 577 #4 11

 

cfs yr in mi ag Q

s d d yr in ft mi ft

3 . 66 3 . / 30 100 ) (

400 , 86 1 365 1 12 1 2 5280 2

 

 

cfs MGD d cap gal cap wwtp Q

MGD cfs gal MG

4 . 19 5 . 12 125 000 , 100 ) (

548 . 1 10 1

6

   

 

cfs yr in mi ur Q

s d d yr in ft mi ft

1 . 11 5 . / 30 10 ) (

400 , 86 1 365 1 12 1 2 5280 2

 

Solution to loading problem (cont.)

 CBOD5 loading

David A. Reckhow CEE 577 #4 12

d lb d mi lb mi ag W 4000 40 100 ) (

2 2

       

 

d lb L mg MGD wwtp W

L mg MGD d lb

2810 15 . / 180 5 . 12 ) (

/ * / 34 . 8

 

 

d lb L mg cfs d lb L mg cfs ur W 1620 / / 4 . 5 / 27 1 . 11 ) (   

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CEE 572 Lecture #4 9/21/2017 7

Solution to loading problem (cont.)

 Lead loading

David A. Reckhow CEE 577 #4 13

d lb d mi lb mi atm W 13 1 . 3 . 1 100 ) (

2 2

       

 

d lb L g MGD wwtp W

g mg L mg MGD d lb

11 . 05 . / 22 5 . 12 ) (

3

10 / * / 34 . 8

 





 

d lb g mg L mg cfs d lb L g cfs ur W 8 . 16 10 / / 4 . 5 / 280 1 . 11 ) (

3

          



 

Other Terms in the Mass Balance

 Outflow  Reaction  Settling

David A. Reckhow CEE 577 #4 14

Outflow Qc 

Reaction kM kVc  

Settling vA c k Vc

s s

 

k v H

s 

V A H

s



Since: J=vc As

Sediment- water interface

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CEE 572 Lecture #4 9/21/2017 8

Combining all terms:

 Dependent variable: c  Independent variable: t  Forcing function: W(t), the way in which the

external world “forces” the system

 Parameters: V, Q, k, v, As

David A. Reckhow CEE 577 #4 15

V dc dt W t Qc kVc vA c

s

    ( )

Steady State Case

 Mass Balance  Solution  Assimilation factor

 Where  The assimilation or “cleansing” factor

David A. Reckhow CEE 577 #4 16

c W Q kV vAs   

a W c 

s

vA kV Q a   

r

V dc dt W t Qc kVc vA c

s

     ( )

W

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CEE 572 Lecture #4 9/21/2017 9

Steady State Example

David A. Reckhow CEE 577 #4 17

A lake has the following characteristics:

Volume m d C 



50 000

3 1

, Mean Depth = 2 m Inflow = Outflow = 7500 m Temperature = 25

3

The lake receives the input of a pollutant from three sources:

a factory discharge of 50 kg d-1, a flux from the atmosphere

f 0.6 g m-2 d-1, and the inflow stream that has a

concentration of 10 mg/L. If the pollutant decays at the rate

f 0.25/d at 20oC (note: Ɵ=1.05).
a. compute the assimilation factor
b. steady state concentration
c. show breakdown for each term

#3.1 from Chapra

(pg.52)

Example 3.1: Solution

David A. Reckhow CEE 577 #4 18

k d   

  

0 25 0 25 105 0 319

25 20 25 20 1

. . ( . ) . 

First correct the decay rate for temperature Now the assimilation factor

1 3

454 , 23 ) 000 , 50 ( 319 . 7500



     d m kV Q a

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CEE 572 Lecture #4 9/21/2017 10

Example 3.1: Solution (cont.)

David A. Reckhow CEE 577 #4 19

The surface area of the lake is:

A V H m

s 

  50 000 2 25 000

2

, ,

The atmospheric and inflow load is then:

W JA g d

atmosphere s

   0 6 25 000 15 000 . ( , ) , /

W g d

low inf

( ) , /   7500 10 75 000

Combining all loads: W

W W W g d

factory atmosphere low

      

inf

, , , , / 50 000 15 000 75 000 140 000

Example 3.1: Solution (cont.)

David A. Reckhow CEE 577 #4 20

And finally, the concentration:

L mg d m d g a W c / 97 . 5 / 454 , 23 / 000 , 140

3

  

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CEE 572 Lecture #4 9/21/2017 11

Transfer function & residence time

David A. Reckhow CEE 577 #4 21

c W Q kV vA Qc Q kV vA

s in s

      c c Q Q kV vA

in s

    

Transfer function

E E dE dt  w V Q  c

s s

Vc Qc kVc vA c V Q kV vA      

Residence times

generic water contaminant

 Pittsburgh

David A. Reckhow CEE 577 #4 22

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David A. Reckhow CEE 577 #4 27

Bromide in PA

Kelly D. Good and Jeanne

M. VanBriesen, 2016

“Current and Potential Future Bromide Loads from Coal-Fired Power Plants in the Allegheny River Basin and Their Effects on Downstream Concentrations”, ES&T 50, 9078

David A. Reckhow CEE 577 #4 28

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CEE 572 Lecture #4 9/21/2017 15

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 To next lecture

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